STOP
Early Journal Content on JSTOR, Free to Anyone in the World
This article is one of nearly 500,000 scholarly works digitized and made freely available to everyone in
the world by JSTOR.
Known as the Early Journal Content, this set of works include research articles, news, letters, and other
writings published in more than 200 of the oldest leading academic journals. The works date from the
mid-seventeenth to the early twentieth centuries.
We encourage people to read and share the Early Journal Content openly and to tell others that this
resource exists. People may post this content online or redistribute in any way for non-commercial
purposes.
Read more about Early Journal Content at http://about.jstor.org/participate-jstor/individuals/early-
journal-content .
JSTOR is a digital library of academic journals, books, and primary source objects. JSTOR helps people
discover, use, and build upon a wide range of content through a powerful research and teaching
platform, and preserves this content for future generations. JSTOR is part of ITHAKA, a not-for-profit
organization that also includes Ithaka S+R and Portico. For more information about JSTOR, please
contact support@jstor.org.
CRITICISMS AND DISCUSSIONS. 309
To bring these magical squares to the surface the squares of
each set of parallel squares may be permuted as follows:
Original order 1, 2, 3, 4, 5, 6,
Permuted order 3, 2, 1, 6, 5, 4.
The result is the final cube shown in the beginning of this article.
The above permutation is subject to two conditions. The sev-
eral sets of parallel squares must all be permuted in the same man-
ner. Any two parallel squares which in the original cube are located
on opposite sides of the middle plane of the cube and at an equal
distance from it, in the permuted cube must be located on opposite
sides of the middle plane of the cube and at an equal distance from it.
These conditions are for the protection of the diagonals.
John Worthington.
MAGIC IN THE FOURTH DIMENSION.
Definition of terms : Row is a general term ; rank denotes a hori-
zontal right-to-left row; file a row from front to back; and column
a vertical row in a cube — not used of any horizontal dimension.
If n 2 numbers of a given series can be grouped so as to form a
magic square and n such squares be so placed as to constitute a
magic cube, why may we not go a step further and group n cubes
in relations of the fourth dimension ? In a magic square containing
the natural series 1 . . . w 2 the summation is— — ' ; in a magic cube
fl( fl^-i— T ^
with the series i...» 3 it is — - — ^— -; and in an analogous fourth-
dimension construction it naturally will be — - — — — .
With this idea in mind I have made some experiments, and the
results are interesting. The analogy with squares and cubes is not
perfect, for rows of numbers can be arranged side by side to repre-
sent a visible square, squares can be piled one upon another to make
a visible cube, but cubes cannot be so combined in drawing as to
picture to the eye their higher relations. My expectation a priori
was that some connection or relation, probably through some form
of diagonal-of-diagonal, would be found to exist between the cubes
containing the n* terms of a series. This particular feature did ap-
pear in the cases where n was odd. Here is how it worked out:
I. When n is odd.
1. Let w=3, then 8=123. — The natural series 1. . .81 was di-
vided into three sub-series such that the sum of each would be
3io
THE MONIST.
one-third the sum of the whole. In dealing with any such series
when n is odd there will be n sub-series, each starting with one of
the first w numbers, and the difference between successive terms will
be w+i, except after a multiple of n, when the difference is i. In
the present case the three sub-series begin respectively with i, 2, 3,
and the first is 1 5 9 10 14 18 19 23 27 28 32 36 37 41 45 46 50 54
55 59 63 64 68 72 73 yy 81. These numbers were arranged in
three squares constituting a magic cube, and the row of squares
so formed was flanked on right and left by similar rows formed from
the other two sub-series (see Fig. 1).
It is not easy — perhaps it is not possible — to make an abso-
lutely perfect cube of 3. These are not perfect, yet they have many
25
38
60
28
77
18
67
8
48
33
79
11
72
1
SO
21
40
62
65
6
52
23
45
55
35
75
13
29
78
16
68
9
46
26
39
58
70
2
5i
19
41
63
31
80
12
24
43
56
36
73
14
66
4
53
69
7
47
27
37
59
30
76
17
20
42
61
32
81
10
71
3
49
34
74
15
64
5
54
22
44
57
Fig. 1. (3')
striking features. Taking the three cubes separately we find that
in each all the "straight" dimensions — rank, file and column — have
the proper footing, 123. In the middle cube there are two plane
diagonals having the same summation, and in cubes I and III one
each. In cube II four cubic diagonals and four diagonals of vertical
squares are correct ; I and III each have one cubic diagonal and one
vertical-square diagonal.
So much for the original cubes; now for some combinations.
The three squares on the diagonal running down from left to right
will make a magic cube with rank, file, column, cubic diagonals,
two plane diagonals and four vertical-square diagonals (37 in all)
correct. Two other cubes can be formed by starting with the top
squares of II and III respectively and following the "broken diag-
CRITICISMS AND DISCUSSIONS.
3"
onals" running downward to the right. In each of these S occurs
at least 28 times (in 9 ranks, 9 files, 9 columns and one cubic diag-
onal). Various other combinations may be found by taking the
squares together in horizontal rows and noting how some columns
and assorted diagonals have the proper summation, but the most
important and significant are those already pointed out. In all the
sum 123 occurs over 200 times in this small figure.
317
473
604
10
161
T92
348
479
510
36
67
223
354
385
536
567
98
229
260
4"
442
598
104
135
286
110
■36
292
448
579
6to
n
167
323
454
485
5"
42
198
329
360
386
542
73
204
235
261
4'7
573
79
423
554
85
236
267
298
429
585
ITI
1+2
'73
304
460
61 1
■ 7
48
179
335
486
517
548
54
210
361
392
21 r
367
398
529
60
86
242
273
404
560
586
"7
148
279
435
461
6'7
23
154
3'0
336
492
523
29
185
504
35
186
342
498
379
535
6i
217
373
254
410
561
92
248
129
285
436
592
'23
4
160
3"
467
623
606
12
168
324
455
481
512
43
'99
330
356
387
543
74
205
231
262
418
574
80
106
137
293
449
580
299
430
58l
112
'43
(74
305
456
612
18
49
180
33'
487
5'8
549
55
206
362
393
424
555
81
237
268
87
243
274
405
556
587
118
149
280
43'
462
618
24
'55
306
337
493
524
30
181
212
368
399
530
56
380
531
62
218
374
255
406
562
93
249
130
281
437
593
124
5
'56
3'2
468
624
505
3'
'87
343
499
■93
349
480
506
37
68
224
355
38>
537
568.
99
230
256
412
443
599
105
131
287
3'8
474
605
6
162
'75
3t"
457
613
'9
50
176
332
488
5'9
550
5'
207
363
394
425
55'
82
238
269
300
426
582
"3
'44
588
l'9
150
276
432
463
619
25
151
307
338
494
525
26
182
213
309
400
526
57
88
244
275
40J
557
25 r
407
563
94
250
126
282
438
594
125
1 ,
'57
3'3
469
625
501
32
188
344
500
376
532
63
219
375
38
€9
225
351
382
538
569
100
226
257
4U
444
600
101
132
288
319
475
601
7
■63
'94
350
476
507
482
5J3
44
200
326
357
388
544
75
201
232
263
419
575
76
107
138
294
450
576
607
13
169
325
45'
464
620
21
rS2
308
339
495
521
27
183
214
370
396
527
58
89
245
271
402
558
589
120
146
277
433
127
283
439
595
121
2
158
J'4
470
621
502
33
189
345
496
377
533
64
220
37'
252
408
564
95
246
570
96
227
258
414
445
596
102
'33
289
320
47'
602
8
164
'95
346
477
508
39
7P
221
352
383
539
358
389
545
71
202
233
264
420
57'
77
108
'39
295
446
577
608
'4
'70
321
452
483
5'4
45
196
327
46
177
333
489
520
546
52
208
364
395
421
552
83
239
270
296
427
583
"4
'45
171
302
458
614
20
3
159
3J5
466
622
503
34
190
34'
497
378
534
65
216
372
253
409
565
9'
247
128
284
440
59'
122
441
597
'03
134
290
3.6
472
603
9
'65
191
347
478
509
40
66
222
353
384
540
566
97
288
259
4'5
234
265
416
572
78
109
140
291
447
578
609
'5
166
322
453
484
5'5
4'
197
328
359
390
54'
72
203
547
53
209
365
391
422
553
84
240
266
297
428
584
"5
14'
172
303
459
615
16
47
■ 78
334
490
516
340
491
5«
28
■84
215
366
397
528
59
90
241
272
403
559
590
116
'47
278
434
465
616
22
'53
30S
Fig. 2. (s 4 )
One most interesting fact remains to be noticed. While the
three cubes were constructed separately and independently the figure
formed by combining them is an absolutely perfect square of 9, with
a summation of 369 in rank, file and corner diagonal (besides all
"broken" diagonals running downward to the right), and a perfect
312 THE MONIST.
balancing of complementary numbers about the center. Any such
pair, taken with the central number 41, gives us the familiar sum 123,
and this serves to bind the whole together in a remarkable manner.
2. Let w=5, then 8=1565. — In Fig. 2 is represented a group
of 5-cubes each made up of the numbers in a sub-series of the nat-
ural series 1 . . .625. In accordance with the principle stated in a pre-
vious paragraph the central sub-series is 1 7 13 19 25 26 32 ... 625,
and the other four can easily be discovered by inspection. Each of
the twenty-five small squares has the summation 1565 in rank, file,
corner diagonal and broken diagonals, twenty times altogether in
each square, or 500 times for all.
Combining the five squares in col. I we have a cube in which
all the 75 "straight" rows (rank, file and vertical column), all the
horizontal diagonals and three of the four cubic diagonals foot up
1565. In cube III all the cubic diagonals are correct. Each cube
also has seven vertical-square diagonals with the same summation.
Taking together the squares in horizontal rows we find certain
diagonals having the same sum, but the columns do not. The five
squares in either diagonal of the large square, however, combine to
produce almost perfect cubes, with rank, file, column and cubic
diagonals all correct, and many diagonals of vertical squares.
A still more remarkable fact is that the squares in the broken
diagonals running in either direction also combine to produce cubes
as nearly perfect as those first considered. Indeed, the great square
seems to be an enlarged copy of the small squares, and where the
cells in the small ones unite to produce S the corresponding squares
in the large figure unite to produce cubes more or less perfect.
Many other combinations are discoverable, but these are sufficient
to illustrate the principle, and show the interrelations of the cubes
and their constituent squares. The summation 1565 occurs in this
figure not less than 1400 times.
The plane figure containing the five cubes (or twenty-five
squares) is itself a perfect square with a summation of 7825
for every rank, file, corner or broken diagonal. Furthermore all
complementary pairs are balanced about the center, as in Fig. 1.
Any square group of four, nine or sixteen of the small squares is
magic, and if the group of nine is taken at the center it is "perfect."
It is worthy of notice that all the powers of « above the first lie in
the middle rank of squares, and that all other multiples of « are
grouped in regular relations in the other ranks and have the same
CRITICISMS AND DISCUSSIONS.
313
grouping in all the squares of any given rank. The same is true
of the figure illustrating 7*, which is to be considered next.
3. Let n=7, then 8=8407. — This is so similar in all its prop-
erties to the 5-construction just discussed that it hardly needs sep-
arate description. It is more nearly perfect in all its parts than the
5*, having a larger proportion of its vertical-square diagonals cor-
rect. Any square group of four, nine, sixteen, twenty-five or thirty-
six small squares is magic, and if the group of nine or twenty-five
I
255
254
4
248
10
11
245
240
18
19
237
25
231
230
28
252
6
7
249
13
243
242
16
21
235
234
24
228
30
31
225
8
250
251
5
241
15
14
244
233
23
22
236
32
226
227
29
253
3
2
256
12
246
247
9
20
238
239
17
229
27
26
232
224
34
35
221
4i
215
214
44
49
207
206
52
200
58
59
197
37
219
218
40
212
46
47
209
204
54
55
201
61
195
194
64
217
39
38
220
48
210
211
45
56
202
203
53
193
63
62
196
36
222
223
33
213
43
42
216
205
51
50
208
60
198
199
57
192
66
67
189
73
183
182
76
81
175
174
84
168
90
91
165
69
187
186
72
180
78
79
177
172
86
87
169
93
163
162
96
185
71
70
188
80
178
179
77
88
170
171
85
161
95
94
164
68
190
191
65
181
75
74
184
173
83
82
176
92
166
167
89
97
159
158
100
152
106
107
149
144
114
115
141
121
135
134
124
156
102
103
153
109
147
146
112
«7
139
138
120
132
126
127
129
104
154
155
101
145
iti
1 10
148
137
119
118
140
128
130
131
125
157
99
98
160
108
150
151
105
116
142
143]
113
133
123
122
136
Fig. 3- (4 4 )
be taken at the center of the figure it is "perfect." The grouping
of multiples and powers of n is very similar to that already described
for 5*.
II. When n is even.
I. Let «=4, then 8=514. — The numbers may be arranged in
either of two ways. If we take the diagram for the 4-cube as
II
III
1
1295
1294
3
1292
6
1278
20
21
1276
23
"73
37
1259
1258
39
1256
42
1290
8
■288
1287
11
7
25
1271
27
28
1268
1272
'254
44
.1252
1251
47
43
1284
1283
'5
16
>4
1279
31
32
1264
1263
1265
36
1248
1247
5'
52
50
'243
'3
'7
1281
1282
1280
18
1266
1262
34
33
35
1261
49
53
1245
1246
1244
54
12
1286
9
10
1289
1285
1267
29
1270
1269
26
30
48
1250
45
46
1253
1249
1291
3
4
'293
5
1296
24
1277
■275
22
•274
'9
1255
38
40
'257
4'
1260
1188
no
in
1 186
"3
1 183
127
1 169
1168
129
1166
'32
"52
146
'47
"50
'49
"47
"5
1 181
"7
118
1 178
1 182
1 164
134
1 162
1 161
'37
'33
'5'
"45
'53
154
1142
"46
121
122
1 174
"73
"75
126
1158
"67
141
142
140
"53
'57
■58
1 138
"37
"39
162
1176
1 172
124
123
125
1171
■39
'43
"55
"56
"54
144
1 140
"36
160
'59
161
"35
1177
119
1180
1 179
116
120
'38
u6o
'35
•36
"63
"59
1 141
'55
"44
"43
152
•56
114
1187
1185
112
1184
I09
1165
128
130
1167
'3'
1 170
'5°
"5'
"49
148
1 148
'45
217
1079
1078
219
1076
222
1062
236
237
1060
239
1057
253
1043
1042
255
1040
258
1074
224
1072
1071
227
223
241
i°55
243
244
1052
1056
1038
260
1036
1035
263
259
1068
1067
231
232
230
1063
247
248
1048
1047
1049
252
1032
1031
267
268
266
1027
229
233
1065
1066
1064
234
1050
1046
250
249
251
'045
265
269
1029
1030
1028
270
228
1070
225
226
1073
I069
105 1
245
1054
">53
242
246
264
1034
261
262
1037
>°33
1075
218
220
1077
221
1080
240
106 1
1059
238
1058
235
1039
254
256
1041
257
1044
865
43t
43°
867
428
870
4H
884
885
4U
887
409
901
395
394
903
392
906
426
872
424
423
875
871
889
407
891
892
404
408
390
908
388
387
9"
907
420
419
879
880
878
415
895
896
400
399
401
900
384
&S
915
916
914
379
877
881
417
418
416
882
402
398
898
897
899
397
9'3
9'7
381
382
380
918
876
422
873
874
425
421
403
'893
406
405
890
894
912
386
909
910
389
385
427
866
868
429
869
432
888
4'3
411
886
410
883
39'-
902
904
393
9°5
396
864
434
435
862
437
859
451
845
844
453
842
456
828
470
47'
826
473
823
439
857
441
442
854
858
840
458
838
837
461
457
475
821
477
478
818
822
445
446
850
849
8S«
450
834
833
465
466
464
829
481
482
814
8>3
815
486
852
848
448
447
449
847
463
467
83'
832
830
468
816
812
484
483
485
811
853
443
856
855
440
444
462
836
459
460
839
835
8i7
479
820
819
476
480
438
863
861
436
860
433
841
452
454
843
455
846
474
827
825
472
824
469
756
542
543
754
545
751
559
737
736
56i
734
564
720
578
579
7'8
58i
7'5
547
749
549
55°
746
750
732
566
73°
729
569
565
583
7'3
585
586
710
7'4
553
554
742
741
743
558
726
725
573
574
572
721
589
59°
706
7°5
707
594
744
740
556
555
557
739
57"
575
723
724
722
576
708
704
592
59'
593
703
745
55"
748
747
548
552
57°
728
567
568
73"
727
709
587
712
7"
584
588
546
755-
753
544
752
541
733
560
562
735
563
738
582
7W
7'7
580
7J6
577
Fig. 4, First Part. (6*: 8=3891)
IV
VI
122$
71
70
1227
68
1230
1224
74
75
1222
77
1219
120^
92
93
1204
95
1201
r>r>
1232
64
f>3
'235
■23'
79
1217
81
82
1214
1218
97
"99
99
100
119C
1200
fio
59
'239
1240
1238
55
85
86
1210
1209
[211
90
>°3
104
1192
1191
"93
108
'237
124!
57
58
56
1242
1212
1208
88
87
89
1207
"94
1190
106
■°5
107
1189
1236
62
<m
1234
65
61
1213
83
1216
1215
80
84
"95
101
1 198
"97
98
102
(>7
1226
1228
69
1229
72
78
1223
1221
7*
1220
73
96
1205
1203
94
1202
91
180
1 1 18
1 1 19
178
1121
'75
181
'"5
1 114
'83
1 1 12
186
'99
1097
1096
20 r
IO94
204
1123
'73
1125
I 126
170
'74
11 10
188
1108
1107
KJI
187
1092
206
1090
1089
209
205
1129
1130
166
"•'5
167
"34
1 104
1103
'95
196
194
1099
1086
1085
213
214
212
1081
168
164
1132
"3'
"33
""'3
'93
'97
MOI
1 102
MOO
198
211
2'5
1083
1084
I082
216
169
1127
172
171
1124
1128
192
1 106
189
190
1 109
"05
210
1088
207
208
I09I
1087
1122
'79
'77
1120
176
1 1 17
1111
182
184
"'3
185
1 1 16
1093
200
202
1095
203
1098
1009
287
286
1011
284
1014
ico8
290
291
1006
293
1003
990
308
309
988
3"
985
282
1016
280
279
1019
1015
295
IOOI
297
298
998
1002
3'3
983
3'5
3'6
980
984
276
275
1023
1024
1022
271
30'
302
994
993
995
306
3'9
320
976
975
977
324
102 1
1025
273
274
272
1026
996
992
3°4
303
305
99'
978
974
322
321
323
973
1020
278
1017
1018
281
'77
997
299
1000
999
296
300
979
3'7
982
981
3'4
3'8
283
1010
1012
285
1013
288
294
1007
1005
292
1004
289
312
989
987
3'0
986
3°7
S<"
935
934
363
932
366
360
938
939
358
94'
355
342
956
957
340
959
337
93°
368
928
927
371
367
943
353
945
946
350
354
961
335
963
964
332
336
924
923
375
376
374
919
949
95°
346
345
347
954
967
968
328
327
329
972
373
377
921
922
920
378
348
344
952
95'
953
343
330
326
970
969
97'
325
372
926
369
370
929
925
349
947
352
35'
944
948
33'
965
334
333
962
966
93'
362
3 6 4
933
365
936
942
359
357
940
356
937
960
34'
339
958
338
955
504
794
795
502
797
499
505
79'
790
5°7
788
510
523
773
772
525
77°
528
799
497
801
802
494
498
786
5'2
784
783
5'5
5"
768
53o
766
765
533
529
805
806
490
489
491
810
780
779
5'9
520
5'8
775
762
76i
537
538
536
757
492
488
808
807
809
487
5'7
52'
777
778
776
522
535
539
759
760
758
540
493
803
496
495
800
804
5'6
782
5'3
5'4
785
781
534
764
53'
532
767
763
798
503
5°'
70
500
79i
787
506
508
789
509
792
769
524
526
771
527
774
612
686
687
610
689
607
613
683
682
6'5
680
618
631
66 S
664
633
662
636
691
605
693
694
602
606
678
620
676
675
623
619
660
638
658
657
641
637
697
698
598
597
599
702
672
671
627
628
626
667
654
653
645
646
644
649
£00
595
700
699
701
595
625
629
669
670
668
630
643
647
651
652
650
648
601
695
604
603
692
696
624
674
621
622
677
673
642
656
639
640
659
655
690
611
609
688
608
685
679
614
616
681
617
684
661
632
634
663
635
666
Fig. 4, Second Part. (6':S=389i)
316
THE MONIST.
given in Magic Squares and Cubes and simply extend it to cover
the larger numbers involved we shall have a group of four cubes
in which all the "straight" dimensions have S=SI4, but no diag-
onals except the four cubic diagonals. Each horizontal row of
squares will produce a cube having exactly the same properties as
those in the four vertical rows. If the four squares in either diag-
I
409S
4094
4
5
4091
4090
8
4032
66
67
4029
4028
70
71
4025
4088
10
11
4085
4084
14
IS
4081
73
4023
4022
76
77
4019
4018
80
4080
18
19
4077
4076
22
23
4073
81
4015
4014
84
85
401 1
4010
88
25
4071
4070
28
29
4067
4066
32
4008
90
91
4005
4004
94
95
4001
406S
31
30
4068
4069
27
26
4072
96
4002
4003
93
92
4006
4007
89
24
4074
4075
31
20
4078
4079
17
4009
87
86
4012
4013
83
82
4016
16
4082
4083
13
12
4086
4087
9
4017
79
78
4020
4021
75
74
4024
4089
7
6
4092
4093
3
2
4096
72
4026
4027
69
68
4030
4031
65
4064
34
35
4061
4060
38
39
4057
97
3999
3098
100
101
399S
3994
104
41
4055
4054
44
45
4051
4050
48
3992
106
107
3989
3988
1 10
III
3985
49
4047
4046
52
53
4043
4042
56
3984
114
115
398i
3980
118
119
3977
4040
58
59
4037
4036
62
63
4033
121
3975
3974
124
125
3971
3970
128
64
4034
4035
61
60
4038
4039
57
3969
127
126
3972
3973
123
122
3976
4041
55
54
4044
4045
51
50
4048
120
3978
3979
117
116
3982
3983
113
4049
47
46
4052
4053
43
42
4056
112
3986
3987
109
108
3990
3991
ios
40
4058
4059
37
36
4062
4063
33
3993
103
ro2
3996
3997
99
98
4000
II IV
Fig. 5, 8*, First Part (One cube written).
onal of the figure be piled together neither vertical columns nor
cubic diagonals will have the correct summation, but all the diagonals
of vertical squares in either direction will. Regarding the whole
group of sixteen squares as a plane square we find it magic, having
the summation 2056 in every rank, file and corner diagonal, 1028
CRITICISMS AND DISCUSSIONS.
317
in each half-rank or half-file, and 514 in each quarter-rank or
quarter-file. Furthermore all complementary pairs are balanced about
the center.
The alternative arrangement shown in Fig. 3 makes each of the
small squares perfect in itself, with every rank, file and corner diag-
onal footing up 514 and complementary pairs balanced about the
3968
130
131
3965
39&t
134
135
396i
193
3903
3902
196
«97
3899
3898
200
137
3959
3958
140
141
3955
3954
144
3896
202
203
3893
3892
206
207
3889
145
3051
3950
148
149
3947
3946
152
3888
210
211
3885
3884
214
215
3881
3944
154
155
3941
3940
158
IS9
3937
217
3879
3878
220
221
3875
3874
224
160
3938
3939
157
156
3942
3943
153
3873
223
222
3876
3877
219
218
3880
3945
151
150
3948
3949
147
146
3952
216
38R?
3883
213
212
3886
3887
209
3953
143
142
3956
3957
139
138
3960
208
3890
3891
205
204
3894
3895
201
136
3962
3963
133
132
3966
3967
129
3897
199
198
3900
3901
195
194
3904
161
3935
3934
164
165
3931
393°
168
3872
226
227
3869
3868
230
231
3865
3928
170
171
3925
3924
»74
175
3921
233
3863
3862
236
237
3859
38S8
240
3920
178
179
35W7
39i6
182
183
3913
241
3855
3854
244
245
3851
3850
248
185
39"
39«o
188
«89
3907
3906
192
.1848
250
251
3845
3844
254
255
3841
3905
191
190
3908
3909
187
186
3912
256
3842
3843
253
252
3846
3847
249
184
3914
3915
181
180
3918
3919
177
3849
247
246
3852
3853
243
242
3856
176
3922
3923
173
172
3926
3927
169
3857
239
238
3860
3861
335
234
3864
3929
167
166
3932
3933
163
162
3936
232
3866
3867
229
228
3870
3871
225
VI VIII
Fig. 5, 8*, Second Part (One cube written).
center. As in the other arrangement the squares in each vertical
or horizontal row combine to make cubes whose "straight" dimen-
sions all have the right summation. In addition the new form has
the two plane diagonals of each original square (eight for each
cube), but sacrifices the four cubic diagonals in each cube. In lieu
318 THE MONIST.
of these we find a complete set of "bent diagonals" ("Franklin")
like those described for the magic cube of six in The Monist for
July, 1909.
If the four squares in either diagonal of the large figure be
piled up it will be found that neither cubic diagonal nor vertical
column is correct, but that all diagonals of vertical squares facing
toward front or back are. Taken as a plane figure the whole group
makes up a magic square of 16 with the summation 2056 in every
rank, file or corner diagonal, half that summation in half of each
of those dimensions, and one-fourth of it in each quarter dimension.
2. Let n=6, then 8=3891. — With the natural series 1. . .1296
squares were constructed which combined to produce the six magic
cubes of six indicated by the Roman numerals in Fig. 4. These
have all the characteristics of the 6-cube described in The Monist
of July last — 108 "straight" rows, 12 plane diagonals and 24 "bent"
diagonals in each cube, with the addition of 32 vertical-square
diagonals if the squares are piled in a certain order. A seventh
cube with the same features is made by combining the squares in
the lowest horizontal row — i. e., the bottom squares of the num-
bered cubes. The feature of the cubic bent diagonals is found on
combining any three of the small squares, no matter in what order
they are taken. In view of the recent discussion of this cube it seems
unnecessary to give any further account of it now.
The whole figure, made up as it is of thirty-six magic squares,
is itself a magic square of 36 with the proper summation (23346)
for every rank, file and corner diagonal, and the corresponding
fractional part of that for each half, third or sixth of those dimen-
sions. Any square group of four, nine, sixteen or twenty-five of the
small squares will be magic in all its dimensions.
3. Let n=8, then 8=16388. — The numbers 1...4096 may be
arranged in several different ways. If the diagrams in Mr. An-
drews's book be adopted we have a group of eight cubes in which
rank, file, column and cubic diagonal are correct (and in which
the halves of these dimensions have the half summation), but all
plane diagonals are irregular. If the plan be adopted of construct-
ing the small squares of complementary couplets, as in the 6-cube,
the plane diagonals are equalized at the cost of certain other features.
I have used therefore a plan which combines to some extent the ad-
vantages of both the others.
It will be noticed that each of the small squares in Fig. 5 is
CRITICISMS AND DISCUSSIONS. 319
perfect in that it has the summation 16388 for rank, file and corner
diagonal (also for broken diagonals if each of the separated parts
contain two, four or six — not an odd number of cells) , and in balan-
cing complementary couplets. When the eight squares are piled
one upon the other a cube results in which rank, file, column, the
plane diagonals of each horizontal square, the four ordinary cubic
diagonals and 32 cubic bent diagonals all have S=i6388. What is
still more remarkable, the half of each of the "straight" dimensions
and of each cubic diagonal has half that sum. Indeed this cube of
eight can be sliced into eight cubes of 4 in each of which every rank,
file, column and cubic diagonal has the footing 8194; and each of
these 4-cubes can be subdivided into eight tiny 2-cubes in each of
which the eight numbers foot up 16388.
So much for the features of the single cube here presented.
As a matter of fact only the one cube has actually been written out.
The plan of its construction, however, is so simple and the relations
of numbers so uniform in the powers of 8 that it was easy to in-
vestigate the properties of the whole 8 4 scheme without having the
squares actually before me. I give here the initial number of each
of the eight squares in each of the eight cubes, leaving it for some
one possessed of more leisure to write them all out and verify my
statements as to the intercubical features. It should be remembered
that in each square the number diagonally opposite the one here
given is its complement, i. e., the number which added to it will
give the sum 4097.
I
II
III
IV
V
VI
VII
VIII
I
3840
3584
769
3072
1281
1537
2304
4064
289
545
3296
1057
2784
2528
1825
4032
321
577
3264
1089
2752
2496
1857
97
3744
3488
865
2976
1377
1633
2208
3968
385
641
3200
1 153
2688
2432
1921
161
3680
3424
929
2912
1441
1697
2144
193
3648
3392
961
2880
1473
1729
2112
3872
481
737
3104
1249
2592
2336
2017
16388 16388 16388 16388 16388 16388 16388 16388
Each of the sixty-four numbers given above will be at the
upper left-hand corner of a square and its complement at the lower
right-hand corner. The footings given are for these initial numbers,
320 THE MONIST.
but the arrangement of numbers in the squares is such that the
footing will be the same for every one of the sixty-four columns
in each cube. If the numbers in each horizontal line of the table
above be added they will be found to have the same sum: conse-
quently the squares headed by them must make a cube as nearly
perfect as the example given in Fig. 5, which is cube I of the table
above. But the sum of half the numbers in each line is half of
16388, and hence each of the eight cubes formed by taking the
squares in the horizontal rows is capable of subdivision into 4-cubes
and 2-cubes, like our original cube. We thus have sixteen cubes,
each with the characteristics described for the one presented in
Fig- 5-
If we pile the squares lying in the diagonal of our great square
(starting with 1, 289, etc., or 2304, 2528, etc.) we find that its col-
umns and cubic diagonals are not correct; but all the diagonals of
its vertical squares are so, and even here the remarkable feature of
the half-dimension persists.
Of course there is nothing to prevent one's going still further
and examining constructions involving the fifth or even higher pow-
ers, but the utility of such research may well be doubted. The purpose
of this article is to suggest in sketch rather than to discuss exhaus-
tively an interesting field of study for some one who may have time
to develop it.
H. M. Kingery.
Wabash College.